Finite Group Databasebeta

← #11740256

Subgroups — #11740256

Order 896.

644 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 201 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×C22⋊Q8C211yesmaximal, characteristic
2C2×#6507C211yesmaximal, characteristic, complemented
3C2×#6487C211yesmaximal, characteristic
4C22×C4×Dic7C211yesmaximal, characteristic
5#90002C211yesmaximal, characteristic, complemented
6#89748C211yesmaximal, characteristic, complemented
7#89746C211yesmaximal, characteristic, complemented
8#89744C214yesmaximal, complemented
9#89714C214yesmaximal
10C22×C7×Q8C2211yescharacteristic
11C23×C4×C7C2211yescharacteristic
12C23×Dic7C2212yes—
13C22×Dic14C2211yescharacteristic
14C7×C22⋊Q8—24no—
15C2×C7×C4⋊C4C2212yes—
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2212yes—
18C2×C22⋊Dic7C2212yes—
19#6507—24no—
20C2×C4⋊Dic7C2211yescharacteristic
21#6487—24no—
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2214yes—
24C2×Dic7⋊C4C2214yes—
25C2×C4×Dic7—24no—
26C2×C4×Dic7C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28C2×C4×Dic7C2211yescharacteristic
29C7⋊#95C2214yes—
30C7⋊#95C2214yes—
31#2857—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C2×C7×Q8—22no—
34C2×C7×Q8—22no—
35C2×C7×Q8—22no—
36C22×C4×C7—22no—
37C22×C4×C7—22no—
38C22×C4×C7D414yes—
39C22×C4×C7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×Dic7—24no—
44C22×Dic7—24no—
45C22×Dic7Q818yescomplemented
46C22×Dic7C2312yes—
47C22×Dic7C2312yes—
48C22×Dic7C2312yes—
49C22×Dic7C2312yes—
50C2×Dic14—22no—
51C2×Dic14—22no—
52C2×Dic14—22no—
53C7×C4⋊C4—22no—
54C7×C4⋊C4—24no—
55C7×C22⋊C4—24no—
56C22⋊Dic7—24no—
57C4⋊Dic7—22no—
58Dic7⋊C4—24no—
59Dic7⋊C4—28no—
60Dic7⋊C4—28no—
61C4×Dic7—22no—
62C4×Dic7—24no—
63C4×Dic7D414yes—
64C2×C4⋊Q8—71no—
65C2×C22⋊Q8—71no—
66C2×C22⋊Q8D711yescharacteristic
67C22×C42—71no—
68#391—71no—
69#391—71no—
70#389—74no—
71#384—71no—
72#380—74no—
73C23×C7—21no—
74C23×C7—22no—
75C23×C7—21no—
76C23×C7C2×D412yes—
77C23×C7C2×Q814yes—
78C23×C7C2411yescharacteristic
79C7×Q8—44no—
80C2×C4×C7—21no—
81C2×C4×C7—21no—
82C2×C4×C7—28no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—24no—
87C2×C4×C7—22no—
88C2×C4×C7C2×D412yes—
89C2×C4×C7C2×D412yes—
90C2×C4×C7C2×D412yes—
91C2×C4×C7#4112yes—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—216no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—24no—
99C2×Dic7—22no—
100C2×Dic7C2×D414yes—
101C2×Dic7C2×Q814yes—
102C2×Dic7C2×Q814yes—
103C2×Dic7#4114yes—
104Dic14—44no—
105C22×Q8—71no—
106C22×Q8C2×D711yescharacteristic
107C23×C4—72no—
108C23×C4C2×D711yescharacteristic
109C4⋊Q8—144no—
110C22⋊Q8—24no—
111C22⋊Q8—144no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4—71no—
114C2×C4⋊C4—74no—
115C2×C4⋊C4—74no—
116C2×C4⋊C4C2×D711yescharacteristic
117C2×C4⋊C4C2×D712yes—
118C2×C22⋊C4—72no—
119C2×C22⋊C4C2×D712yes—
120C2×C42—71no—
121C2×C42—72no—
122C2×C42—71no—
123C2×C42—144no—
124#95—74no—
125#95—74no—
126C22×C7—22no—
127C22×C7—24no—
128C22×C7—22no—
129C22×C7—24no—
130C22×C7C4⋊Q814yescomplemented
131C22×C7C22×D411yescharacteristic
132C22×C7C22×D411yescharacteristic
133C22×C7C22×Q812yes—
134C22×C7C2×#4111yescharacteristic
135C22×C7C2×#4112yes—
136C4×C7—22no—
137C4×C7—42no—
138C4×C7—42no—
139C4×C7#12114yes—
140Dic7—24no—
141Dic7—42no—
142Dic7—42no—
143Dic7C22⋊Q818yescomplemented
144C24C22×D711yescharacteristic
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—142no—
149C2×Q8—142no—
150C2×Q8—142no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—72no—
154C22×C4—78no—
155C22×C4—72no—
156C22×C4—72no—
157C22×C4—72no—
158C22×C4—144no—
159C22×C4—144no—
160C22×C4C22⋊D714yes—
161C22×C4C22×D712yes—
162C22×C4C22×D712yes—
163C22×C4C22×D711yescharacteristic
164C22×C4C22×D711yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—24no—
167C4⋊C4—148no—
168C4⋊C4—148no—
169C4⋊C4—144no—
170C4⋊C4—142no—
171C22⋊C4—24no—
172C22⋊C4—144no—
173C42—74no—
174C42—144no—
175C42—142no—
176C2×C7—24no—
177C2×C7C2×#12111yescharacteristic
178C2×C7C2×C22⋊Q812yes—
179C2×C7C2×C4⋊Q811yescharacteristic
180C2×C7#53211yescharacteristic
181C2×C7#53312yes—
182C23—21no—
183C23—21no—
184C23—22no—
185C23Q8×D714yes—
186C23C2×C22⋊D712yes—
187C23C23×D711yescharacteristic
188Q8—44no—
189Q8—284no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—74no—
199C2×C4—74no—
200C2×C4—74no—
201C2×C4—74no—
202C2×C4—1416no—
203C2×C4—142no—
204C2×C4—142no—
205C2×C4—142no—
206C2×C4—142no—
207C2×C4—144no—
208C2×C4—142no—
209C2×C4—142no—
210C2×C4D4×D712yes—
211C2×C4#115912yes—
212C2×C4C2×C22⋊D712yes—
213C2×C4C2×C22⋊D712yes—
214C7#285711yescharacteristic, complemented, Sylow
215C22—22no—
216C22—22no—
217C22—24no—
218C22—24no—
219C22#650714yescomplemented
220C22C2×D4×D711yescharacteristic
221C22C2×#115911yescharacteristic
222C22C2×Q8×D712yes—
223C22D7×#4112yes—
224C22C22×C22⋊D711yescharacteristic
225C4—22no—
226C4—42no—
227C4—42no—
228C4—78no—
229C4—144no—
230C4—282no—
231C4—282no—
232C4#650214yes—
233C2—24no—
234C2#9031312yes—
235C2D7×C22⋊Q812yes—
236C2C2×#650211yescharacteristic
237C2C2×#650711yescharacteristic
238C2#9052311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C22⋊Q8C2
prime radicalC24×C7C23
hypercenterC2×C22⋊Q8D7
hypocenterC7#2857
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#6507C2semidirect
5. #90002C2semidirect
6. #89748C2semidirect
7. #89746C2semidirect
8. #89744C2semidirect
45. C22×Dic7Q8semidirect
130. C22×C7C4⋊Q8semidirect
143. Dic7C22⋊Q8semidirect
214. C7#2857semidirect
219. C22#6507semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C7×C22⋊Q82
C2×#65072
C2×#64872
C22×C4×Dic72
#900022
#897482
#897462
#897442
#897142
#28577